POWER SYSTEM STATE ESTIMATION

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POWER SYSTEM STATE ESTIMATION

BY

© KAMINI KOUSAL Y A SEKARAN, B.E.

A thesis submitted to the school of Graduate Studies in partial fulfillment of the requirements for the

degree of Master of Engineering

Faculty of Engineering and Applied Science Memorial University of Newfoundland

September 2008

StJohn's Newfoundland Canada

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ABSTRACT

Monitoring and control of a complex interconnected power ystem requires the accurate estimate of its states. Different meters are placed at the variou substations and the measurements are transmitted to the central control center. However, it is likely that there may be etTors associated with the measurements and in some cases, some measurements may not be available. Power system state estimation is a technique by which the state of a power system (usually magnitude and angle of bus voltages) is determined using raw measurements. The results of the state estimation are used for real- time security analysis, optimal power flow, etc. These are also used in calculating the line power flows between buses, by which the system operators will be able to conclude if any line is overloaded, and then take necessary action to prevent any mishap from happening. In the initial part of this thesis, State Estimation (SE) based on Weighted Least Squares (WLS) technique and bad data detection, identification, and elimination are presented. The bad data detection and identification are facilitated by the chi-squared test and normal residual methods. The WLS, chi-Squared test and normal residual methods are implemented in Matlab and tested using different power system models.

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Case studies demonstrate that the WLS technique is reliable in estimating state variable of a power system. Chi-squared and Normal residual methods detect and identify bad data efficiently.

In the second part of this thesis, the transmission line reactance parameter error estimation is formulated and implemented using the residual sensitivity analysis for various power system models. The estimation involves two steps: the eJTOr identification and estimation of the parameter implemented using Matlab. The identification of the parameter error is facilitated by the normalized residual technique, and the parameter error estimation is facilitated using the residual sensitivity analysis. The parameters estimated are reliable and close to the actual (true) value.

In the final part of this thesis the measurements considered to be available for state estimation are a few synchronized phasor measurements in addition to the conventional measurement data, to enhance the performance of the state estimator, for a very large power system. The phasor measurement, when present in sufficient numbers, with other measurements, improves the accuracy of the SE. Matlab is used to implement multi-area SE using various case studies.

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ACKNOWLEDGEMENTS

I would like to thank my supervisor Dr. Benjamin Jeyasurya for his constant advice, guidance and encouragement during all stages of this research.

Special thanks and appreciation are given to the Natural Sciences and Engineering Research Council of Canada and to Memorial University of Newfoundland for the financial support that made this research possible. Thanks are also given to the Faculty of Engineering at Memorial University for providing the resources to carry out this research.

Finally, I would also like to express my deepest gratitude, to my mentor and my father Mr. K. Sekaran, my mother Mrs. Lalitha Sekaran and my husband Mr. Kathiravan Chenthilnathan for their constant encouragement, motivation, understanding and support.

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LIST OF ABBREVIATIONS

SE State Estimation

AGC Automatic Generation Control WLS Weighted Least Square EMS Energy Management System RMU Remote Terminal Unit PMU Phasor Measurement Unit GPS Global Positioning System

SCDR Symmetrical Component Distance Relay CSE Central State Estimator

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LIST OF FIGURES

2.1 Energy control center system security schematic ... 10

2.2 Flowchart for Weighted Least Square Technique ... 17

2.3 Two-p01t n-model of a network branch ... 18

2.4 6-bus network ... 24

2.5 Comparison between actual and estimated values of the bus voltage magnitude, in KV ... 26

2.6 Comparison between actual and estimated values of the bus phase angle, in degree ... 26

2.7 Comparison between actual and estimated bus real (Pi) power injection, in MW ... 27

2.8 Compruison between actual and estimated bus reactive (Qi) power injection, in MVAR ... 27

2.9 Comparison between actual and estimated real line power injection, in MW (Pijl-2 is the real line power flow from bus 1 to bus 2) ... 28

2.10 Comparison between actual and estimated reactive line power flow, in MV AR (Qij1-2 is the real line power flow from bus 1 to bus 2) ... 28

2.11 10 unit 39 bus New England test system ... 30

2.12 Comparison between a few actual and estimated bus voltage magnitudes, in KV ... 32

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2.13 Comparison between a few actual and estimated bus phase angles, in degree .... 32 2.14 Comparison between a few actual and estimated (Pi) real bus power injections, in MW ... 33 2.15 Comparison between a few actual and estimated (Qi) reactive bus power

injections, in MY AR ... 33 2.16 Comparison between a few actual and estimated real line power flows, in MW

(Pij7-6 is the real line power flow from bus 7 to bus 6) ... 34 2.17 Comparison between a few actual and estimated reactive line power flows, in MVAR (Qij2-1 is the real line power flow from bus 2 to bus 1) ... 34

3.1 Measurements in a power system ... 37 3.2 Flow chart showing the Chi-square method to detect bad data ... 41 3.3 Flow chart showing the Largest Residual Test to identify and eliminate bad

data ... 47 3.4 Comparison between few real power injections calculated with four bad data and

after the elimination of bad data with the actual value, in MW ... 50 3.5 Comparison between few real line power flows calculated with four bad data and after the elimination of bad data with the actual value, in MW ... 53

4.1 Shows the measurements included in the parameter estimation ... 64 4.2 Comparison between actual and estimated parameter values for6-bus system with one parameter error. ... 67

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4.3 Comparison between actual and estimated parameter values for 6-bus system with

two parameter errors ... 68

4.4 Comparison between actual and estimated parameter values for 39-bus system with one parameter error. ... 69

4.5 Comparison between actual and estimated values parameter values for 39-bus system with two parameter errors ... 70

4.6 Comparison between actual and estimated parameter values for 39-bus system with four parameter errors ... 71

5.1 Phasor Measurement Unit (PMU) ... 77

5.2 Bus assignment for areas ... 82

5.3 Data and measurement exchange ... 85

5.4 Control areas of 39-bus system ... 87

5.5 Comparison between the voltage magnitudes of actual, integrated SE & two-level SE value for the 39-bus system ... 88

5.6 Comparison between the real line power flow of actual, integrated SE & two-level SE value for the 39-bus system ... 89

5.7 Comparison between the reactive line power flow of actual, integrated SE & two- level SE value for the 39-bus System ... 89

5.8 IEEE 118-bus system ... 93

5.9 Control areas of the IEEE 118-bus system ... 94

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- - - · -·--- -- - - -

5.10 Compatison between the voltage magnitude of actual, integrated SE & two-level SE value for 100% loading condition of the 118-bus system ... 96 5.11 Comparison between the real line power flow of actual, integrated SE & two-level SE value for 100% loading condition of the ll8-bus system ... 96 5.12 Comparison between the reactive line power flow of actual, integrated SE & two- level SE value for 100% loading condition of the ll8-bus system ... 97 5.13 Comparison between the voltage magnitude of actual, integrated SE & two-level SE value for 75% loading condition of the ll8-bus system ... 97 5.14 Comparison between the real line power flow of actual, integrated SE & two-level SE value for 75% loading condition of the 118-bus system ... 98 5.15 Comparison between the reactive line power flow of actual, integrated SE & two- level SE value for 75% loading condition of the 118-bus system ... 98 5.16 Comparison between the voltage magnitude of actual, integrated SE & two-level SE value for 50% loading condition of the 118-bus system ... 99 5.17 Comparison between the real line power flow of actual, integrated SE & two-level SE value for 50% loading condition of the 118-bus system ... 99 5.18 Comparison between the reactive line power flow of actual, integrated SE & two-

level SE value for 50% loading condition of the 118-bus system ... 100

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LIST OF TABLES

3.1 The bad data introduced to the 6-bus system and

l(x)

value ... 43 3.2 The bad data introduced to the 39-bus system and

l(x)

value ... 44 3.3 The normalized residual for the 6-bus system ... 48 3.4 The value

l(x)

calculated for the 6-bus system after elimination of bad data ... 49 3.5 The normalized residual for the 39-bus system ... 51 3.6 The

l(x)

value calculated after the elimination of bad data for the 39-bus

System ... 52

4.1 The normalized residuals for the 6-bus system with one parameter error

(for the line 3-5) ... 63 4.2 The normalized residual values after parameter correction for the 6-bus system with one parameter error (for the line 3-5) ... 66 4.3 The estimated parameter values for 6-bus system with one parameter error. ... 66 4.4 The estimated parameter values for 6-bus system with two parameter errors .... 68 4.5 The estimated parameter values for the 39 bus system with one parameter

error. ... 69 4.6 The estimated parameter values for the 39-bus system with two parameter

errors ... 70

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4.7 The estimated parameter values for 39-bus system with four parameter

en·ors ... 71

5.1 Measurements for the 6-bus system ... 79

5.2 Estimated state variables for the 6-bus system ... 81

5.3 Estimated line power flows for the 6-bus system ... 81

5.4

l(x )

value calculated for the Area 1 of the 39-bus system ... 91

5.5 Normalized residual values calculated for the CSE of the 39-bus system ... 91

5.6

l(x )

value calculated for the Area 6 of the ll8-bus system ... 101

5.7 Normalized residual values calculated for the CSE of the 118-bus system ... 101

A.l Main characteristic of the 6-bus system ... ll4 A.2 Line characteristics of 6-bus system ... ll4 A.3 Actual and estimated state variables for the 18 and 62 measurement data ... ll4 A.4 Actual and estimated power injections for the 18 and 62 measurement data ... ll5 A.5 Actual and estimated power flows for the 18 and 62 measurement data ... 115

B.1 Main characteristic of the 39-bus system ... ll7 B.2 Line characteristics of the 10-unit 39-bus New England test system ... ll8 B.3 Actual and estimated state variables for the 131 and 277 measurement data .... 119

B.4 Actual and estimated power injections for the 131 and 277 measurement data .. 120

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C.1 Bus distribution for 39-bus system ... 123

C.2 Measurements available for the areas 1 & 2 of 39-bus system ... 123

C.3 Synchronized phasor measurements ... 123

C.4 Result of integrated and multi-area SE Solution ... 123

C.5 Main characteristic of the 118-Bus system ... 126

C.6 Bus distribution for 118-bus system ... 126

C.7 Measurements available for the areas 1-9 of 118-bus system ... 125

C.8 Synchronized phasor measurements ... 125

C.9 Result of integrated and multi-area SE solution ... 125

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CHAPTER!

INTRODUCTION

1.1 OVERVIEW OF POWER SYSTEM STATE ESTIMATION

The importance of State Estimation (SE) in an electrical power system can be realized by considering the North-east blackout 2003 that took place in US and Canada, due to poor control-room procedures and failure of the power grid organization to keep it

from spreading [1]. This shows that a better power system estimating tool is necessary for the safe operation of the power system and it can be established by the implementation of state estimation in power systems.

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State estimation gained popularity in the 1950's and 1960's as it was used in the military industry to derive estimates of the trajectory of a missile, plane or spacecraft based on a redundant set of imperfect measurements usually based on radar tracking of it position and velocity vector [2]. The static state estimation for a power system network based on the power flow model was first proposed by Fred Schweppe [3].

State estimation is a technique developed to provide an estimate of an unknown system state variable and to quantitatively analyze the estimated state vatiable before it is used for real-time power-flow calculations or on-line system security assessment. A state estimator is a data processing algorithm for converting redundant meter readings and other available information into an estimate of the state of an electric power system. It plays an essential part in every energy management system and also is a basic tool in ensuring the secure operation of a power system

State estimation is the process of assigning a value to an unknown system state variable based on measurements from that system according to certain ctiteria. This process involves imperfect measurements that are redundant being used in estimating the system states, based on a statistical criterion that estimates the true value of the state vruiables to minimize or maximize the selected criterion [2]. The most widely used criterion is that of minimizing the sum of the squares of the differences between the estimated and "true" (i.e., measured) values of a function.

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In a power system, measurements are required in order to estimate the system performance in real time for both system secu1ity control and constraints on economic dispatch. The estimator is designed to produce the "best estimate" of the system state variables, recognizing that there are e1Tors in the measured quantities and that there may be redundant measurements. The output data are then used in system control centers in the implementation of the security-constrained dispatch and control of the power system [2].

Transducers for power system measurements, like any measurement device, will be subject to error. If the errors are small, they may go unnoticed or undetected and can cause misinterpretation by those reading the measured values. The instruments may have gross measurement errors that render their output useless. Finally, the telemetry equipment often experiences petiods when communication channels are completely out, thus, depriving the system operator of any information about some parts of the power system network. Power system state estimation techniques have been developed for these reasons. A state estimator can "smooth out" small random errors in the meter's readings, detect and identify gross measurement errors, and "fill in" meter readings that have failed due to communications failures [2]. In general, state estimation is the art of estimating the exact system state given a set of imperfect measurements made on the power system.

Of the many ctiteria that have been examined and used in the literature , the Weighted Least-Square (WLS) technique, where the objective is to minimize the sum of

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the squares of the weighted deviations of the estimated measurements from the actual measurements, is used in this thesis to implement state estimation for various case studies.

1.2 OBJECTIVES OF THE THESIS

For vanous power system models, power system state estimation, bad data processing technique, network parameter error processmg technique and for a large power system addition of phasor measurements to the conventional measurement data are considered in this thesis. A reliable estimate of the system states of any electrical power system is vital for all post state estimation applications and for the effective operation of the power utility. The main goals of this research are as follows:

• To gain thorough understanding of power system state estimation and its features.

• To implement WLS technique, bad data elimination, and finally estimate the network parameter error.

• To enhance the performance of state estimation with the phasor measurements.

• To implement the studied methods for standard power system models

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1.3 ORGANIZATION OF THE THESIS

A brief literature review and an overview of the research for the different chapters are discussed in the respective chapters. In Chapter 2, the power system state estimation for the static ac power system is discussed and illustrated using 6-bus and 39-bus systems. The modeling of the WLS (Weighted Least Square) technique, measurement and components are also briefly illustrated. The 6-bus system is solved with two different measurement data. The results of the estimate system states are compared with the actual value to illustrate the accuracy of theSE. TheSE demonstrates its ability to give the best estimate of the system state variables.

Chapter 3 formulates and implements the bad measurement data processmg techniques. Initially a few bad measurements are introduced into the measurement data.

The bad measurement data are detected using the chi-squared test. They are identified using the normalized residual test and finally eliminated from the measurement data. The 6-bus and 39-bus systems are used to illustrate bad data processing techniques.

The network parameter error estimation is devised and implemented in Chapter 4.

The line reactance parameter error is the focus of this chapter. Errors are introduced into the line reactance for 6-bus and 39-bus systems. The identification of the parameter errors

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is facilitated by the normalized residual technique. The parameter en·ors are estimated and the correct line reactance is updated into the system model.

In addition to the conventional measurement data in a very large power system, the performance of SE in the presence of phasor measurements is studied in Chapter 5. In a multi-area power system, the two-level SE is fotmulated using 6-bus and illustrated using 39-bus and 118-bus power system models. The state variables of the individual area are estimated in the first level of the two-level SE. The measurement data for the Central State Estimator (CSE) comprises the estimated state variables of the individual area, a few boundary and external bus measurements and phasor measurements. The CSE estimates the overall state of the system (which is the second level of the two-level SE).

Chapter 6 provides the conclusion of the thesis and enumerates the contributions of this research. Finally, the possible future research in this area is discussed.

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CHAPTER2

POWER SYSTEM STATE ESTIMATION

2.1 INTRODUCTION

As discussed in Chapter 1, the technique that can provide an estimate of the unknown quantities with a few available measurements is known as state estimation. The purpose of this chapter is to briefly study the application of power system state estimation and to implement it using WLS techniques for various power system models. The power

system models used are 6-bus and 39-bus system models. The estimated state variables are compared with the actual value, to prove that a state estimator can give the best estimate of the state of a power system.

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2.2 APPLICATION OF POWER SYSTEM STATE ESTIMATION (SE)

Power system state estimation is the process of reading field measurements and deriving the best guess of the state of a power system. The power system states are the voltage magnitudes and relative phase angles of all buses in the system. The estimated state variables are used to calculate the estimates of the real and reactive power flow between the lines. With the estimated power flows, the operator in a power system utility will have access to the real time information and take necessary measures in case of overloading to avoid blackouts in a power system. In addition, the power system state variables are used in advanced applications, such as security analysis and optimal power flow, implemented in the control centre.

The energy control centre system security schematic in Figure 2.1 illustrates the information flows between the various functions to be executed in an operations control center. The remote terminal unit performs various functions such as updating the system regarding the current status of the power system and encoding measurement transducer outputs and opened/closed status information into digital signals, which are transmitted to the operations center over the communications circuit. The control center can also transmit control information, such as raise/lower commands to generators and open/close commands to circuit breakers/switches. The information approaching the state estimator

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is broken down into the analog measurements and breaker/switch status indications. The state estimator will process all data before being used by other programs, except the analog measurements of generator outputs, which are used directly, by the Automatic Generation Control (AGC) program.

The next important information necessary for the state estimator is the network topology. This is the information that gives the mapping of the transmission lines to the buses and breaker/switch status (open/close). This information is significant since the breaker/switches status in any substation can cause the network topology to change.

Hence, a program is provided to read the telemetered breaker/switch status indications and restructure the electrical model of the system. The updated electiical model of the power transmission system is sent to the state estimator program together with the analog measurements. The output of the state estimator consists of all bus voltage magnitudes, phase angles, power injections and power flows. The bad data is also identified, detected and, if possible, eliminated by the estimator. The output data together with the electrical model developed by the network topology program provides the basis for the economic dispatch program and contingency analysis program [2] and [4].

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G-.

[Y

Remote erminal units in substations

Telemetry

&

Communications equipment

Breaker or SWitch status indications

Network topology program

Updated system electrical model

Analog Measurements

Generator outputs AGC

I

Generation

raiseJlO\Ner Base points and signals participation factors

Economic dispatch

calculation

r--

^Penatty_factor ^~

calculation ~

System Model Description

Display to operator

state

Estimator Power flows, vottages, etc.

Display to operator

Bad data measureme nt Alarms

state Estimator output

Contingency analysis

~ ^algorithm

Generation correctrve action

~ calculation

Pate over I

ntial oad s alarm

Displa y to oper at or Carre ction

eve oad to reli overl

r

Displa y to operator

Figure 2.1: Energy control center sy tern security schematic [2]

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2.3 IMPLEMENATION OF STATE ESTIMATION

The electric power transmission system uses various meters to measure real power, reactive power, bus voltages and currents. The current and potential transfmmers installed in the transmission lines, transformers, buses of the power plant or a substation monitor these continuous or analog quantities. The analog quantities monitored using the transducers are passed on to the analog-to-digital converters. The digital outputs are then telemetered to the energy control center over various communication links. The data received at the energy control center are processed by computers, which infmm the system operators of the present state of the power system.

The acquired data always contains inaccuracies, which are unavoidable as physical measurements cannot be entirely free of random error or noise. These errors can be quantified in a statistical sense and the estimated values of the quantities being measured are then either accepted or rejected if certain measures of accuracy are exceeded. Due to noise, the true values of the physical quantities are never known, so a technique must be used to calculate the best possible estimates of the unknown quantities.

There are various methods used to formulate the best estimate of the unknown parameter. A few of the most commonly used criteria are the maximum likelihood criterion, the weighted least-square and the minimum variance criterion. The one used in

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the following case studies is the weighted least-square criterion, where the objective is to minimize the sum of the squares of the weighted deviations of the estimated measurement from the actual measurement.

2.3.1 Weighted Least Square

Due to noise or random error, the true value of any physical quantity is not known; hence, a suitable procedure has to be followed to calculate the best estimate of the unknown quantity [2]. The method of least squares is often used to "best fit"

measured data relating to two or more quantities. The modeling and implementation of the weighted least-square method is explained using two case studies in the next section.

The ac systems used in the case studies are 6-bus and 39-bus systems. These systems are solved with two different measurement data with error.

2.4 CASE STUDIES

In an ac power system, the measured quantities are power injections, line power flows, currents, transformer tap position and voltage magnitude. The function of the measurement equations and their partial derivatives (Jacobian matrix) are all nonlinear.

Therefore, solving an ac system is complicated. The unknown state variables of an ac

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power system are the bus voltage magnitudes and bus phase angle (except the reference bus). The two systems are solved with two measurement data. The 6-bus system is solved with 18 and 62 measurement data, and the 39-bus system with 131 and 277 measurement data. The state estimator's ability to estimate the state va1iables with a small number of measurements is tested.

2.4.1 Modeling of Weighted Least Square Technique

WLS state estimation minimizes the weighted sum of the squares of the residuals.

Consider the measurement set vector

z

as in equation (2.1):

Zt h₁(x₁,x₂,x₃^{· • ·},x,) et

z2 h₂(x_{1 ,}x_{2 ,}x₃^·^{· ·,}x,.) + ez

=h(x)+e (2.1)

z=

=

Zm h₁₁₁(x₁,x₂,x₃···,x,) em

Z₁₁₁^] is the measurement vector.

h;

(x)

is the nonlinear function relating measurement i to the state vector x

x,.]

is the system state vector

e₁₁₁^] is the vector of the measurement eiTors.

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m is the number of measurements and n is the number of state variables to be estimated.

Let

E(e)

denote the expected value of

e

with the following assumption:

E(e;)

⁼^0, ^{i =}^l,···,m (2.2)

Measurement errors are independent, i.e.

E[ ^eie ^j]

⁼⁰ ^V^j ^t ⁱ^.Hence the covariance of the error is given as:

cov(e) =

E[e.eT]

⁼^R⁼diag{~

,a; , ... a;}

(2.3)

ai

is the standard deviation of each measurement i, calculated to reflect the expected accuracy of the meter used [5].

The objective function to be minimized by weighted least square is given in equation (2.4). It is the square of the difference between each measured value and the true value divided by the covariance of the error.

(2.4)

=

[z- h(x)Y

^R-¹

[z- h(x)]

The gradient of the objective (residual vector) is taken and then equated to zero.

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g(x) = a~~) =

^- ^H^T

^(x ^)R ^-

¹^[^z- ^h(x^)]

⁼

⁰ ^where,

^H(x) ⁼ [a~~)]

^(2.5)

Then, expanding the nonlinear function

g(x)

into Taylor series and neglecting the higher order terms lead to an iterative scheme, as in the Gauss-Newton method shown in equation (2.6), which is used to solve equation (2.5)

(2.6)

where k is the iterative index and xk is the solution vector at the iteration k .

G(x)

is the gain matrix, which is expressed in equation (2.7):

c(xk

)=

^H^T(xk

)R-

¹^H(x^k)

g(xk

) =

^- ^H^r^(x^k

)R- '(z

^{-h(xk ))}

(2.7) (2.8)

Usually, the gain matrix is sparse and is decomposed into triangular factors. The sparse linear set of equations is solved using forward or backward substitution at each iteration k , where ~k+l

=

^xk+l^- ^{xk :}

(2.9)

The iteration is continued until Maxj~k

I < ^c

^where

^c

is a very small value.

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2.4.2 Weighted Least Square Algorithm

The iterative solution to equation (2.9) gtves a reliable WLS estimate of the unknown state variable. The algorithm used for the technique is outlined in the flowchart, shown in Figure 2.2.

1. Begin the iteration by setting the iteration index k

=

0 and defining ku,;1 to any desired value, so that when the solution does not converge, it will stop the iteration. Then, set flat start values 1 and 0 to bus voltage magnitudes and bus phase angles, respectively. Finally, £ is set to a very small value.

2. Terminate the iteration when k > kumit·

3. Calculate the measurement function

h(xk ) ,

the Jacobian matrix

H(xk)

^and

the gain matrix

c(xk ) =

H

T ( xk

^)R-

1

_H

(xk ).

4. Solve

!1

^{X k}using equation (2.9).

5. If

1!1

^X^k

I > c ,

then go to step 2. Otherwise stop (algorithm converged).

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No Convergence

Set the iteration index k =0 and initialize the tate vector xk to the flat start V = 1 &

8

⁼0

et the£ and ktimit

Yes Calculate the measurement

function h(xk )

Calculate the measurement Jacobian matrix

H(x k )

and the Gain matrix

G(xk)

Calculate the tuk using

lc(x k

)~k+l

= H r (xk )R -• l z- h(x k )J

No

Update k = k

+

1

&

xk+I =xk +&k

Figure 2.2: Flowchart for Weighted Least Square Technique

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2.4.3 Measurement and Component Modeling

The measurements in an ac system are mainly of three types, bus power injection, line power flows and bus voltage magnitudes. These quantities can be expressed using the state variables. If there are N buses in a system then there will be (2N-l) state variables comprising N bus voltage magnitudes and N-1 bus phase angles, since at the reference bus, the bus phase angle is zero. Let i and j be the two buses as shown in Figure 2.3, which is an equivalent 1t model of a two bus system.

§j ~

... ^....__..

gsi.+jbsi.

~ ~

^gsj+jbsj

_L _L

-- -^-

Figure 2.3: Two-port n-model of a network branch [5]

The real and reactive power injection at a bus can be expressed as in equations (2.10) and (2.11) respectively.

P.

_l

v

l

. I v .(c ..

J lj

^cos e

l)

.. + B

_lj

. sin e

_lJ

.. )

(2.10)

jE N;

Qi v

I

. I v

J

. (G ..

lj

^sin e.

I)

- B ..

_I)

cos e

_I)

. )

_(2.11)

jE N;

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where

P;

^andQ;are respectively the real and reactive bus power injection at bus i.

V; ,B; is the voltage magnitude and phase angle at bus i and Bij

=

B; - Bj Gij

+

jBij is the ij th element of the complex bus admittance matrix.

N; are the buses that are connected to bus i directly.

The real and reactive power flow from bus ito bus j are expressed in equations (2.12) and (2.13) respectively.

PI'}..

= v

_I²

(g

Sl

. ⁺ g

I}

.. ) - v

I

.v .

}

(g

I}

.. ^cos e

I)

.. ⁺ ^b

I}

^.. ^sin e

I)

.. )

(2.12)

Q ..

_I) = -

v

_I² ⁽_~

b

_Sl

. + b

_I)

.. ) - v

_I

.v .

_J

( g

_I)

.. ^sin e..

_I}

- b

_I}

.. cos e

_I}

.. )

(2.13)

where P;j and Qij are the real and reactive line power flow from bus i to bus j.

gij

+

jbij is the admittance of the series branch connecting buses i andj. g si

+

jbs; is the admittance of the shunt branch connected at bus i.

The formulae for the Jacobian of the real power injection measurement are given in equations (214)-(2.17).

a _{:\ e} ^p; _.

⁼

~

_~

^V

_I

^. ^{V .}

_}

^(- ^{G ..}

_I}

^sin ^B

_I)

^.. ⁺ ^{B ..}

_I)

^cos ^{B. )}

_I]

^- ^V

_I²

^{B ..}

_II

U I j = [

(2.14)

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a ^Pi

⁼

v . v . (c .. ⁱⁿ e . - ^B ^. ^co e . . )

a e .

Î ^} Î} Î} Î) Î)

1

(2.15)

aP .

_ _ I -

av

_I

.

~

N

V l G .. cos e .. ⁺ ^B ^.. ^sin e .. ) ⁺ ^V ^. ^{G ..}

L....J }

~ Î) Î) Î} ^!} Î ÎI

j=l

(2.16)

a ^p ^; ⁼ ^V ^. (G .. ^cos e.. ⁺ ^{B ..} ^sin e. )

a v .

Î Î) Î) Î} Î}

1

(2.17)

The formulae for the Jacobian of the reactive power injection measurement are given in equations (2.18)-(2.21)

~

N

^V ^{.V .} (G . ^cos ^B ^.. ⁺ ^{B ..} ^sin e .. ) - ^V

²

G ..

L....J

Î ^} Î} Î} Î} Î) Î ÎI ^(2.18)

j=l

a _ae ^Q ^; _. ⁼ ^V ^{.V .} ^(- ^G ^.. ^cos ^B.. ^- ^{B ..} ^sin ^B ^.. ⁾

I } I} I} I) I)

1

(2.19)

~

N

V . (G .. ^sin e .. - ^{B ..} ^cos e .. ) - ^V ^. ^{B ..}

L....J }

Î} Î} ^!} Î} Î ÎI ^(2.20)

j=l

aQ

ⁱ ^{ ⁾

----'- =

_av _. V

_I

. \G ..

_I)

sin e

₁₎

.. - ^{B ..}

_I}

^cos e

_I)

..

1

(2.21)

The formulae for the Jacobian of real line power flow measurement are given m

equations (2.22)-(2.25).

(41)

(2.22)

(2.23)

aP

ij (

8 ) ( )

_ ___.::.._ = - V

₁ g iJ

cos Bu - bu sin

iJ -

2

g iJ

+

g

s i Vi

av

_I

.

_. ^(2.24)

aP ( )

- -11

'--

= - V .

g ..

cos B + b . sin B ..

a v .

Î ^lj ^lJ Î} Î)

J

(2.25)

The formulae for the Jacobian of the reactive line power flow measurement are given in equations (2.26)-(2.29).

aQ .. ( )

--''-'1 = - v i v j g ^ij

cos

f) ij

+

b ij

sin

f) ij (2.26)

a e.

I

a ^Q

^ij ^{= -} ^v ^j

(g

^ij

^sin

^f)^ij ^-

^b

^ij

^cos

^f)^ij⁾^-

² ^(b

^ij

⁺ ^b

^si^{) vi} ^(2.28)

a

^{v i}

aQ .. ( )

- - 1 ' -

1

= - ^V

^g^..

^{sin B} ^.. ^- ^b ^.. ^{cos B} ^..

^(2.29)

a v

Î Î) ¹⁾ ^lj Î)

j

(42)

The values of the Jacobian for the voltage magnitudes are:

av

I

.

av .

1

^I

· av

₁

. ₀ av

_I

.

· ae

₁

. ⁰

^(2.30)

av

I

The Jacobian matrix

H

has a number of rows and columns equal to the number of measurements and state variables, respectively.

2.4.4 6-Bus System

The six-bus ac network as in Figure 2.4 is the first system of this case study, which is used for the implementation of the state estimation using WLS. There are 6 buses in this system, i.e. N=6. Therefore, 2N-1=11 state variables are to be estimated, which comprises N=6 bus voltage magnitudes and N-1=5 bus phase angles. Bus 1 is the slack bus (the bus phase angle is zero). The buses 2 and 3 represent a bus connected to a generating station (PV buses). Buses 4, 5 and 6 represent a bus connected to the load centre (PQ buses). There are eleven transmission lines and hence there are 44 line power flows (11 *2=22 real and 11 *2=22 reactive). Details of the 6-bus system are given in Appendix A.

This system has a 62 measurement data set that consists of 6 real and 6 reactive bus power injections, 22 real and 22 reactive line power flows and 6 bus voltage

(43)

----- -- - - -

magnitudes. To compare the results and to check the petformance of the state estimation with less number of measurements, the system is initially solved with only 18 measurement data and finally with 62 measurement data. The 18 measurement data consists of only the bus power injections and the bus voltage magnitudes.

Matlab [6] is used to formulate the iterative algorithm for the WLS due to its ability to solve matrices of higher order. Set

c =

^0.0001^and^k

=

0. The matrices

h(xk ) ,

the Jacobian matrix

H(x k )

^and^{the gain}^matrix

G(xk )

are all calculated. The Jacobian

H

matrix for the 18 and 62 measurement data are calculated using the appropriate formulas.

The order of the H matrix for 18 measurement data is 18Xll and for 62 measurement data it is 62X11. The iteration continues until the maximum estimated values

11

X k are

Jess than or equal to the chosen value of the

c, i.e.l~

^{X k}

I ~

^£ ^.If the value is greater

than c, then X k is updated and the iteration continues until the inequality is met. The algorithm converges at the fourth iteration, giving a reliable estimate to the system. Using the estimated state variables, the line power flows are calculated.

(44)

Bus 3

Bus 2 246.1KVL -4.3"

i t-o

o--

1 - - - 1!

:!~ ---~

i 241.5KVL-3.7"

Bus 6

! - - - -

Bus 1 Bus 5

241KVLO" _~ 23l.OKVL-5.9"

o-~

________

_j---~

! - - - -

Bus 4

226.7KVL -5.3"

.o----J

^gene^rator

227.6KVL -4.2"

~load

Figure 2.4: 6-bus network [2].

In order to evaluate the performance of the state estimator, a base case or a reference case of the system is required. Hence, the system is solved using the Power World Simulator [7] and the power flow list from the simulator is assumed to be the actual or true power flow values of this system. The estimated values are compared against the actual values using a bar chart. Different shades are used in the bar chart to represent each case (blank - for actual case, slightly shaded - for 18 measurement data and dark -for 62 measurement data).

(45)

The comparison between estimated state va.Iiables and the actual values for both 18 and 62 measurement data are given in Figure 2.5 and Figure 2.6. The estimated (both real and reactive) bus power injection and line power flows values are also plotted against the actual values, for both 18 and 62 measurement data, as shown in Figure 2.7 through Figure 2.10. In Figure 2.9 and Figure 2.10 only a few randomly chosen line power flows are compared.

In the case of the bus phase angle plot in Figure 2.6, the 62 measurement data provides a closer estimation to the actual value and in bus voltage magnitude plot in Figure 2.5, 18 measurement data provides a closer estimation to the actual value. In the power injection and line power flow plot, the 62 measurement data gives a close estimate to the actual case. By summarizing the results of the estimated values, it can be seen that they are very close to the actual case values and differ only by a few degrees in both cases. This also shows that even with smaller set of measurement data, the state estimation is able to perform well and provide a reliable estimate of state variables.

(46)

Comparison between the actual and estimated values of the voltage magnitudes, in KV

250

;> 245 ::.:::

.5 240 ] aS 235

·~ 230

C':l

5 225

~ 01)

~ 220

~ ₂₁₅ 210

r- r-

f- 1-

--

^{- ' -}

2

r-1-

-

1- !-

0 Actual Value

'"""I-

.-- _{r -} 0 18Measurerrent

f- Data

• 62 Meas urerrent Data

'-'-- ' - ' - ' - l...l...

3 4 5 6

Bus number

Figure 2.5: Comparison between actual and estimated values of the bus voltage magnitude, in KV

0 -1

~

'"'

^-2

01)

~

'0

.5 _-3 Oil aS

c -4

C':l

~

"'

^-5

C':l

.c ~

-6 -7

Comparison between actual and estimated values of the bus phase angles, in degree

0 Actual Value

CJ 18 Measurerrent Data

• 62 Measurerrent Data

2 3 4 5

Bus number

Figure 2.6: Comparison between actual and estimated values of the bus phase angle, in degree

(47)

s:

~

~ 0

-; c.

~

r..

"""

⁰

~ .5

"0

·a a

l:l.ll

~ co:

Comparison between the actual and estimated bus real power injection, in MW

140

120 _r-

--

¹^{- -} ¹^{- -}¹^~-

^--

r-

100 ^1~-^- - J-..- - 1 -

- ·-

80 ,,...._.__ - - ^1- - ⁰Actual Value

.-- .-r- r-

60 -- ^- _.--~ - ^-

.-.- ⁰18 Meas urerren t

40 Data

20 - ⁱ

-

^• 62 Maesurerrent

Data

0 ^{I -} ^L- ^{' - -} ^{' -} ^{I -} ^L..o.

P1 P2 P3 P4 P5 P6

Real

rus

power injection (Magnitude)

Figure 2.7: Comparison between actual and estimated bus real (P;) power injection, in MW

~

r.:

0 ~

c.

~ >

::::::~

co:<

~~

₀

r:::

~

·-

"0

·a a

l:l.ll

~ co:

Comparison between the actual and estimated reactive bus power injection, in MV AR

100 90

80 - - -^{1 - - - -} ^1- ^{- -}

- - -

^- ^...

70 60

r-

r- r - r

l'i 1- 0 Actual Value

F

^(;:

50 40 30 20 10 0

-

^- _f· r,

~ ¹ ^: r.:~

^{0 18 M}easurement

Data

~

^F

n

tiT ^l ^i. ^r ^~~

^I•

^, ^.,

^•62 Measurerrent

I'• I~ ^~"^' ^Data

'-- '-- c...L- I - L-L.;.

Ql Q2 Q3 Q4 Q5 Q6

Reactive

rus

power injection (Magnitude)

Figure 2.8: Comparison between actual and estimated bus reactive (Q;) power injection, in MV AR

(48)

~

⁴⁵⁴⁰

I: _r..~ 35 ^-

~ ~ 30

0 c..

-;; 25

~

r.. 20

... ^-

0

~ 15

"C

.a

₁₀

· s

Oil 5

~ CCI

0

Comparison between the actual and estimated real line power flows, in MW

, - _

_ -

- ₀Actual value

--

0 18 Measurenent

Data

• 62Measurenent

ri ll _{i l i}

^~ ^Data

--

Pij1-2 Pij2-3 Pij3-5 Pij4-5 Pij5-6 Pij6-3 Few real line power flow (Magnitude)

Figure 2.9: Comparison between actual and estimated real line power injection, in MW (Pij 1-2 is the real line power flow from bus 1 to bus 2)

r.."

~ ~ 0 c..

...

~

~ ^~^CCI_~

~

... r..

0 .5

~

"C

·s .a

Oil

~ CCI

Comparison between the actual and estimated reactive line power flows, in MV AR

70 60 50 40 30 20 10 0

Qij1-2 Qij2-3 Qij3-5 Qij4-5 Qij5-6 Qij6-3 Few reacti~ line power flow (Magnitude)

0 Actual value

0 18 Me as urenen t Data

• 62 Measurenent Data

Figure 2.10: Comparison between actual and estimated reactive line power flow, in MV AR (Qij 1-2 is the real line power flow from bus 1 to bus 2)

(49)

2.4.5 39-Bus System

The 10-unit 39 bus New England test ac network as shown in Figure 2.11 is the second system of this case study, which is used for the implementation of the state estimation using WLS. This system is chosen to check the performance of the state estimator with a larger power system. This system is also solved with two measurement data sets, 131 and 277. There are 39 buses in this system, i.e. N=39. Therefore, 2N-1=77 state variables to be estimated, which comprises N=39 bus voltage magnitudes and N- 1=38 bus phase angles. Bus 31 is the slack bus (the bus phase angle is zero). The buses 31, 32, 33, 34, 35, 36, 37, 38 and 39 represent a bus connected to a generating station (PV buses). Buses 3, 4, 7, 8, 12, 15, 16, 18, 20, 21, 23, 24, 25, 26, 27, 28 and 29 represent a bus connected to the load centre (PQ buses). Details of the 39-bus system are given in Appendix B.

There are 34 transmission lines and 12 transformers with tap ratio one, hence there are 184 line power flows (46*2=92 real and 46*2=92 reactive). The 277 measurement data consists of 27 real and 27 reactive bus power injections, 92 real and 92 reactive line power flows and 39 bus voltage magnitudes. Voltage magnitudes, power injections and a few line power flows form the 131 measurement data. The voltage level for the generating station buses and the transmission line buses are 13.8KV and 345KV, respectively

(50)

Matlab is used to formulate the iterative algorithm for the WLS. Set £=0.0001 and k = 0. The mattices

h(xk ) ,

^{the Jacobi}an matrix

H(x k )

and the gain matrix

G(xk )

^are

all calculated. The Jacobian H matrix for both measurement data are calculated using approptiate formulas. The order of the H matrix for 277 measurement data is 277X77 and l31X77 for 131 measurement data. The iteration continues until the maximum

estimated values

!1xk

are less than or equal to the chosen value of the£, i.e.

£ . If the value is greater than£, then X k ts updated and the iteration

continues until the inequality is met.

JS

22

0--J

^generator

1---+

^load

Figure 2.11: 10 unit 39-bus New England test system (8].

(51)

The algorithm converges at the fourth iteration, giving a reliable estimate to the sy tern.

Using the estimated state variables, the line power flows are calculated.

In order to evaluate the petformance of the state estimator, a base case or a reference case of the system is required. Hence, the system is solved using the Power World Simulator and the power flow list from the simulator is assumed to be the actual or true power flow values of this system. The estimated values are compared against the actual values using a bar chart. Different shades are used in the bar chart to represent each case (blank - for actual case, slightly shaded -for 131 measurement data and dark - for 277 measurement data).

This system is quite large; hence only five randomly chosen values are used for comparison. The plot for the comparison between estimated state variables and the actual values are given in Figure 2.12 and Figure 2.13. The estimated (both real and reactive) bus power injection and line power flows values are also plotted against the actual values, as shown in Figure 2.14 through Figure 2.17. From Figure 2.12 and 2.13, it is clear that the estimated values are close to the actual values. The bus power injection plots and the line power flow plots also show that their values lie close to the actual case values. By summarizing the results of the estimated values in the entire figure, it can be seen that they lie close to the actual case values and differ by a slight degree, in only some cases.

This shows that even with smaller measurement data, the state estimation is able to petform well and provide a reliable estimate of state variables.

POWER SYSTEM STATE ESTIMATION

POWER SYSTEM STATE ESTIMATION

BY

© KAMINI KOUSAL Y A SEKARAN, B.E.

ABSTRACT

ACKNOWLEDGEMENTS

LIST OF ABBREVIATIONS

CONTENTS

LIST OF FIGURES

LIST OF TABLES

l(x)

l(x)

l(x)

l(x)

l(x )

l(x )

CHAPTER!

INTRODUCTION

1.1 OVERVIEW OF POWER SYSTEM STATE ESTIMATION

1.2 OBJECTIVES OF THE THESIS

1.3 ORGANIZATION OF THE THESIS

CHAPTER2

POWER SYSTEM STATE ESTIMATION

2.1 INTRODUCTION

2.2 APPLICATION OF POWER SYSTEM STATE ESTIMATION (SE)

G-.

[Y

I

r--

2.3 IMPLEMENATION OF STATE ESTIMATION

2.3.1 Weighted Least Square

2.4 CASE STUDIES

2.4.1 Modeling of Weighted Least Square Technique

z

=

(x)

x,.]

E(e)

e

E(e;)

E[ eie j]

E[e.eT]

,a; , ... a;}

ai

[z- h(x)Y

[z- h(x)]

g(x) = a~~) =

(x )R -

=

H(x) = [a~~)]

g(x)

G(x)

)=

)R-

) =

)R- '(z

=

I < c

c

2.4.2 Weighted Least Square Algorithm

=

h(xk ) ,

H(xk)

c(xk ) =

T ( xk

1

(xk ).

!1

1!1

I > c ,

8

H(x k )

G(xk)

lc(x k

= H r (xk )R -• l z- h(x k )J

+

2.4.3 Measurement and Component Modeling

... ....__..

~ ~

P.

E[ ^eie ^j]

^(x ^)R ^-

⁼

^H(x) ⁼ [a~~)]

I < ^c

^c

... ^....__..

^cos e

^sin e.

. ⁺ g

.. ^cos e

.. ⁺ ^b

^.. ^sin e

.. ^sin e..

a _{:\ e} ^p; _.

^V

^. ^{V .}

^(- ^{G ..}

^sin ^B

^.. ⁺ ^{B ..}

^cos ^{B. )}

^- ^V

^{B ..}

a ^Pi

v . v . (c .. ⁱⁿ e . - ^B ^. ^co e . . )

V l G .. cos e .. ⁺ ^B ^.. ^sin e .. ) ⁺ ^V ^. ^{G ..}

a ^p ^; ⁼ ^V ^. (G .. ^cos e.. ⁺ ^{B ..} ^sin e. )

^V ^{.V .} (G . ^cos ^B ^.. ⁺ ^{B ..} ^sin e .. ) - ^V

a _ae ^Q ^; _. ⁼ ^V ^{.V .} ^(- ^G ^.. ^cos ^B.. ^- ^{B ..} ^sin ^B ^.. ⁾

V . (G .. ^sin e .. - ^{B ..} ^cos e .. ) - ^V ^. ^{B ..}

_av _. V